356 XV NORMED ALGEBRAS AND SPECTRAL THEORY is defined and holomorphic in an open disk | £ | < 1 -f £, for some e > 0, and that F(£) is unitary when )£| = 1. (If |£| = 1, and if we put a = e — x*x, b = e — xx*, y = £e + x, we have e + £x* = &*, yy*~l = £(x + by*~l)9 y*~ly=* {(x + y^a), and finally yy* ~~la — by* ~ {y.) (c) Show that the set of elements x e A such that p(x) ^ 1 is the closed convex hull (Section 12.14, Problem 13) of the set of unitary elements of A. (Use the relation x = no; = — 2rr 6. Let A be a Banach algebra with unit element, endowed with a (not necessarily con- tinuous) involution x\-*x*. Show that the following properties are equivalent: (a) A is a star algebra. (y) \\x* II • IWI = \\x*x\\ for all normal x e A. (8) ||*|| = 1 for all unitary .xe A. (To prove that (8) implies Q8), observe first that (3) implies that A. is a star algebra relative to some norm equivalent to the given norm (Problem 4), and in particular that A is hermitian; then apply Problem 5(c).) 7. TRACES, BITRACES, AND HILBERT ALGEBRAS Let A be an algebra with involution and/a positive linear form on A. In general f(xy) ¥>f(yx) for x, y e A, as example (15.6.6) shows. We say that/is a trace on A if/is a positive linear form satisfying the condition (15.7.1) f(yx)=f(xy) for all x, y in A. For/to be a trace it is sufficient that/(;o:*) =f(x*x) for all x e A; for, replacing x by x + y and using (15.6.2.1), we obtain 0t(f(y*x)) = ^(/(xy*)), and then, replacing x by ix we get S(f(y*x)) = S(f(xy*))9 so that f(y*x) =f(xy*) for all x, y in A. The condition (15.7.1) is trivially satisfied if A is commutative. Example (15.7.2) Let A = :5f(H), where H is a Hilbert space of finite dimension n. n Then the positive linear form f(T) = Y (T - et \ et) is a trace, for it is the i=i n trace Tr(r) = £ £.. of the endomorphism T. On the other hand, it is easily seen a (not necessarily